Learning to Fluctuate: Statistical Foundations for Causal Tabular Pretraining

Zhiheng Zhang

Causal tabular foundation models amortize effect estimation across synthetic mechanisms, but latent-effect supervision rewards posterior shrinkage rather than encoding the repeated-sample response needed in a fixed deployment population. We introduce fluctuation-supervised pretraining (FSP): each synthetic table is labeled by its average treatment effect plus its efficient influence-function fluctuation; deployment remains a frozen forward pass. Along the path $T_{λ,P}=θ(P)+λP_nψ_P$, we prove an endpoint transition: every fixed $λ<1$ retains label ambiguity of order $(1-λ)^2/n$, whereas full fluctuation makes the Gaussian label observable and reduces optimal finite-stratum causal label-prediction risk to order $n^{-2}$. A finite-pretraining bound combines label, network, episode-sampling, and optimization errors; its sampling defect controls fixed-mechanism bias, mean squared error, variance, Gaussian approximation, and, with variance-head accuracy, studentized coverage. Complementary lower bounds separate local $n^{-1}$ ATE risk from the $\log N/M$ excess risk of generic finite-dictionary episode learning. Experiments trace the learned sampling response. Across 24 nonlinear…